TOKYO—When the head of NTT DoCoMo Inc., Japan's biggest mobile carrier, took the stage in January to introduce its latest models, he declared Sony Corp.'s new Xperia Z smartphone his company's top pick, the equivalent of a mother announcing her favorite child.

For Sony, it was a much-needed vote of confidence for one of the company's most important new products in recent memory. The Xperia Z, which went on sale in Japan on Feb. 9, is...